Distinct Fuzzy Subgroups of a Dihedral Group of Order $2pqrs$ for Distinct Primes $p, , q, , r$ and $s$

Document Type : Research Paper


Department of Mathematics, University of Fort Hare, Alice 5700 , Eastern Cape , South Africa


In this paper we classify fuzzy subgroups of the dihedral group $D_{pqrs}$  for distinct primes  $p$, $q$, $r$ and $s$. This follows similar work we have done on distinct fuzzy subgroups of some dihedral groups.
We present formulae for the number of (i) distinct maximal chains of subgroups, (ii) distinct fuzzy subgroups and (iii) non-isomorphic classes of fuzzy subgroups under our chosen equivalence and isomorphism. Some results presented here hold for any dihedral group of order $2n$ where $n$ is a product of any number of distinct primes.


[1] S. Branimir and A. Tepavcevic, A note on a natural equivalence relation on fuzzy power set,
Fuzzy Sets and Systems, 148(2) (2004), 201{210.
[2] C. Degang, J. Jiashang, W. Congxin and E. C. C. Tsang, Some notes on equivalent fuzzy
sets and fuzzy subgroups, Fuzzy Sets and systems, 152(2) (2005), 403{409.
[3] V. Murali and B. B. Makamba, On an equivalence of fuzzy subgroups I, Fuzzy Sets and
Systems 123(2) (2001), 259{264.
[4] O. Ndiweni and B. B. Makamba, Classi cation of fuzzy subgroups of a dihedral group of
order 2pqr for distinct primes p, q and r, International Jounal of Mathematical Sciences and
Engineering Applications, 6(4) (2012) , 159{174.
[5] M. Pruszyriska and M. Dudzicz, On isomorphism between nite chains, Journal of Formalised
Mathematics, 12(1) (2003) , 1{2.
[6] S. Ray, Isomorphic fuzzy groups, Fuzzy Sets and Systems, 50(2) (1992) , 201{207.
[7] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971) , 512{517.
[8] M. Tarnauceanu and L. Bentea, On the number of subgroups of nite abelian groups, Fuzzy
Sets and Systems, 159(10) (2008) , 1084{1096.
[9] A. C. Volf, Counting fuzzy subgroups and chains of subgroups, Fuzzy Systems and Arti cial
Intelligence, 10(3) (2004) , 191{200.